Stability of the surface area preserving mean curvature flow in Euclidean space

TL;DR

This paper proves that in Euclidean space, the surface area preserving mean curvature flow exists indefinitely and converges exponentially to a sphere, given a small initial traceless second fundamental form, even without initial convexity.

Contribution

It establishes long-term existence and exponential convergence of the flow under weaker initial conditions than convexity, focusing on the traceless second fundamental form.

Findings

Flow exists for all time

Flow converges exponentially to a sphere

Convergence holds without initial convexity

Abstract

We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).